Extravagance, irrationality and Diophantine approximation

The paper proves all claimed results with precise hypotheses and rigorous arguments: (i) Bugeaud’s lemma and Denominator/approximation estimates; (ii) an extravagance dichotomy under continued-fraction mixing; (iii) existence of Gauss-invariant ergodic measures with prescribed irrationality exponent; and (iv) a Khinchin-type dichotomy for weak Rényi measures doubling at 0. By contrast, the model’s outline contains critical missteps: (A) a misapplied Borel–Cantelli/Markov bound leading to an unjustified conclusion Y_{n+1}=o(n) in the integrable case, and (B) an unproven uniform comparability of µ(U_q) from 'doubling at 0' plus weak Rényi, and an application of the weak-Rényi Borel–Cantelli lemma to sets not lying in the required σ-field. The conclusions match the paper, but the model’s proof outline is not correct as written.